Theorem dcomex 10203

Description: The Axiom of Dependent Choice implies Infinity, the way we have stated it. Thus, we have Inf+AC implies DC and DC implies Inf, but AC does not imply Inf. (Contributed by Mario Carneiro, 25-Jan-2013.)

Assertion

Ref	Expression
dcomex	⊢ ω ∈ V

Proof of Theorem dcomex

Dummy variables 𝑡 𝑠 𝑥 𝑓 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1n0 8318	. . . . . . 7 ⊢ 1o ≠ ∅
2		df-br 5075	. . . . . . . 8 ⊢ ((𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) ↔ ⟨(𝑓‘𝑛), (𝑓‘suc 𝑛)⟩ ∈ {⟨1o, 1o⟩})
3		elsni 4578	. . . . . . . . 9 ⊢ (⟨(𝑓‘𝑛), (𝑓‘suc 𝑛)⟩ ∈ {⟨1o, 1o⟩} → ⟨(𝑓‘𝑛), (𝑓‘suc 𝑛)⟩ = ⟨1o, 1o⟩)
4		fvex 6787	. . . . . . . . . 10 ⊢ (𝑓‘𝑛) ∈ V
5		fvex 6787	. . . . . . . . . 10 ⊢ (𝑓‘suc 𝑛) ∈ V
6	4, 5	opth1 5390	. . . . . . . . 9 ⊢ (⟨(𝑓‘𝑛), (𝑓‘suc 𝑛)⟩ = ⟨1o, 1o⟩ → (𝑓‘𝑛) = 1o)
7	3, 6	syl 17	. . . . . . . 8 ⊢ (⟨(𝑓‘𝑛), (𝑓‘suc 𝑛)⟩ ∈ {⟨1o, 1o⟩} → (𝑓‘𝑛) = 1o)
8	2, 7	sylbi 216	. . . . . . 7 ⊢ ((𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → (𝑓‘𝑛) = 1o)
9		tz6.12i 6800	. . . . . . 7 ⊢ (1o ≠ ∅ → ((𝑓‘𝑛) = 1o → 𝑛𝑓1o))
10	1, 8, 9	mpsyl 68	. . . . . 6 ⊢ ((𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → 𝑛𝑓1o)
11		vex 3436	. . . . . . 7 ⊢ 𝑛 ∈ V
12		1oex 8307	. . . . . . 7 ⊢ 1o ∈ V
13	11, 12	breldm 5817	. . . . . 6 ⊢ (𝑛𝑓1o → 𝑛 ∈ dom 𝑓)
14	10, 13	syl 17	. . . . 5 ⊢ ((𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → 𝑛 ∈ dom 𝑓)
15	14	ralimi 3087	. . . 4 ⊢ (∀𝑛 ∈ ω (𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → ∀𝑛 ∈ ω 𝑛 ∈ dom 𝑓)
16		dfss3 3909	. . . 4 ⊢ (ω ⊆ dom 𝑓 ↔ ∀𝑛 ∈ ω 𝑛 ∈ dom 𝑓)
17	15, 16	sylibr 233	. . 3 ⊢ (∀𝑛 ∈ ω (𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → ω ⊆ dom 𝑓)
18		vex 3436	. . . . 5 ⊢ 𝑓 ∈ V
19	18	dmex 7758	. . . 4 ⊢ dom 𝑓 ∈ V
20	19	ssex 5245	. . 3 ⊢ (ω ⊆ dom 𝑓 → ω ∈ V)
21	17, 20	syl 17	. 2 ⊢ (∀𝑛 ∈ ω (𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛) → ω ∈ V)
22		snex 5354	. . 3 ⊢ {⟨1o, 1o⟩} ∈ V
23	12, 12	fvsn 7053	. . . . . . . 8 ⊢ ({⟨1o, 1o⟩}‘1o) = 1o
24	12, 12	funsn 6487	. . . . . . . . 9 ⊢ Fun {⟨1o, 1o⟩}
25	12	snid 4597	. . . . . . . . . 10 ⊢ 1o ∈ {1o}
26	12	dmsnop 6119	. . . . . . . . . 10 ⊢ dom {⟨1o, 1o⟩} = {1o}
27	25, 26	eleqtrri 2838	. . . . . . . . 9 ⊢ 1o ∈ dom {⟨1o, 1o⟩}
28		funbrfvb 6824	. . . . . . . . 9 ⊢ ((Fun {⟨1o, 1o⟩} ∧ 1o ∈ dom {⟨1o, 1o⟩}) → (({⟨1o, 1o⟩}‘1o) = 1o ↔ 1o{⟨1o, 1o⟩}1o))
29	24, 27, 28	mp2an 689	. . . . . . . 8 ⊢ (({⟨1o, 1o⟩}‘1o) = 1o ↔ 1o{⟨1o, 1o⟩}1o)
30	23, 29	mpbi 229	. . . . . . 7 ⊢ 1o{⟨1o, 1o⟩}1o
31		breq12 5079	. . . . . . . 8 ⊢ ((𝑠 = 1o ∧ 𝑡 = 1o) → (𝑠{⟨1o, 1o⟩}𝑡 ↔ 1o{⟨1o, 1o⟩}1o))
32	12, 12, 31	spc2ev 3546	. . . . . . 7 ⊢ (1o{⟨1o, 1o⟩}1o → ∃𝑠∃𝑡 𝑠{⟨1o, 1o⟩}𝑡)
33	30, 32	ax-mp 5	. . . . . 6 ⊢ ∃𝑠∃𝑡 𝑠{⟨1o, 1o⟩}𝑡
34		breq 5076	. . . . . . 7 ⊢ (𝑥 = {⟨1o, 1o⟩} → (𝑠𝑥𝑡 ↔ 𝑠{⟨1o, 1o⟩}𝑡))
35	34	2exbidv 1927	. . . . . 6 ⊢ (𝑥 = {⟨1o, 1o⟩} → (∃𝑠∃𝑡 𝑠𝑥𝑡 ↔ ∃𝑠∃𝑡 𝑠{⟨1o, 1o⟩}𝑡))
36	33, 35	mpbiri 257	. . . . 5 ⊢ (𝑥 = {⟨1o, 1o⟩} → ∃𝑠∃𝑡 𝑠𝑥𝑡)
37		ssid 3943	. . . . . . 7 ⊢ {1o} ⊆ {1o}
38	12	rnsnop 6127	. . . . . . 7 ⊢ ran {⟨1o, 1o⟩} = {1o}
39	37, 38, 26	3sstr4i 3964	. . . . . 6 ⊢ ran {⟨1o, 1o⟩} ⊆ dom {⟨1o, 1o⟩}
40		rneq 5845	. . . . . . 7 ⊢ (𝑥 = {⟨1o, 1o⟩} → ran 𝑥 = ran {⟨1o, 1o⟩})
41		dmeq 5812	. . . . . . 7 ⊢ (𝑥 = {⟨1o, 1o⟩} → dom 𝑥 = dom {⟨1o, 1o⟩})
42	40, 41	sseq12d 3954	. . . . . 6 ⊢ (𝑥 = {⟨1o, 1o⟩} → (ran 𝑥 ⊆ dom 𝑥 ↔ ran {⟨1o, 1o⟩} ⊆ dom {⟨1o, 1o⟩}))
43	39, 42	mpbiri 257	. . . . 5 ⊢ (𝑥 = {⟨1o, 1o⟩} → ran 𝑥 ⊆ dom 𝑥)
44		pm5.5 362	. . . . 5 ⊢ ((∃𝑠∃𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → (((∃𝑠∃𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)) ↔ ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)))
45	36, 43, 44	syl2anc 584	. . . 4 ⊢ (𝑥 = {⟨1o, 1o⟩} → (((∃𝑠∃𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)) ↔ ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)))
46		breq 5076	. . . . . 6 ⊢ (𝑥 = {⟨1o, 1o⟩} → ((𝑓‘𝑛)𝑥(𝑓‘suc 𝑛) ↔ (𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛)))
47	46	ralbidv 3112	. . . . 5 ⊢ (𝑥 = {⟨1o, 1o⟩} → (∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛) ↔ ∀𝑛 ∈ ω (𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛)))
48	47	exbidv 1924	. . . 4 ⊢ (𝑥 = {⟨1o, 1o⟩} → (∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛) ↔ ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛)))
49	45, 48	bitrd 278	. . 3 ⊢ (𝑥 = {⟨1o, 1o⟩} → (((∃𝑠∃𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛)) ↔ ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛)))
50		ax-dc 10202	. . 3 ⊢ ((∃𝑠∃𝑡 𝑠𝑥𝑡 ∧ ran 𝑥 ⊆ dom 𝑥) → ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛)𝑥(𝑓‘suc 𝑛))
51	22, 49, 50	vtocl 3498	. 2 ⊢ ∃𝑓∀𝑛 ∈ ω (𝑓‘𝑛){⟨1o, 1o⟩} (𝑓‘suc 𝑛)
52	21, 51	exlimiiv 1934	1 ⊢ ω ∈ V

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  Vcvv 3432   ⊆ wss 3887  ∅c0 4256  {csn 4561  ⟨cop 4567   class class class wbr 5074  dom cdm 5589  ran crn 5590  suc csuc 6268  Fun wfun 6427  ‘cfv 6433  ωcom 7712  1_oc1o 8290

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588  ax-dc 10202

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-fv 6441  df-1o 8297

This theorem is referenced by:  axdc2lem  10204  axdc3lem  10206  axdc4lem  10211  axcclem  10213